A class of angelic sequential non-Fréchet–Urysohn topological groups

Chasco, M.J.; Martín-Peinador, Elena; Tarieladze, V.

doi:10.1016/j.topol.2006.08.008

Cited by 18 publications

(18 citation statements)

References 13 publications

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“…Note that a metrizable lcs E is normable if and only if the strong dual E ′ of E is a Fréchet-Urysohn lcs by [8,Theorem 2.8] (this result can be derived also from [5]). Proof.…”

Section: Applications To Free (Abelian) Topological Groups and Topolomentioning

confidence: 99%

On topological spaces and topological groups with certain local countable networks

Gabriyelyan

Ka̧kol

2015

Topology and its Applications

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The strong Pytkeev property Small base Baire space Topological group Function space (Free) locally convex space Free (abelian) topological groupBeing motivated by the study of the space C c (X) of all continuous real-valued functions on a Tychonoff space X with the compact-open topology, we introduced in [16] the concepts of a cp-network and a cn-network (at a point x) in X. In the present paper we describe the topology of X admitting a countable cp-or cn-network at a point x ∈ X. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [44]. We show that a Baire topological group G is metrizable if and only if G has the strong Pytkeev property. We prove also that a topological group G has a countable cp-network if and only if G is separable and has a countable cp-network at the unit. As an application we show, among the others, that the space D (Ω) of distributions over open Ω ⊆ R n has a countable cp-network, which essentially improves the well known fact stating that D (Ω) has countable tightness. We show that, if X is an MK ω -space, then the free topological group F (X) and the free locally convex space L(X) have a countable cp-network. We prove that a topological vector space E is p-normed (for some 0 < p ≤ 1) if and only if E is Fréchet-Urysohn and admits a fundamental sequence of bounded sets.

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“…Note that a metrizable lcs E is normable if and only if the strong dual E ′ of E is a Fréchet-Urysohn lcs by [8,Theorem 2.8] (this result can be derived also from [5]). Proof.…”

Section: Applications To Free (Abelian) Topological Groups and Topolomentioning

confidence: 99%

On topological spaces and topological groups with certain local countable networks

Gabriyelyan

Ka̧kol

2015

Topology and its Applications

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“…Recall also that a topological space X is strongly Fréchet-Urysohn if for every x ∈ X and for each decreasing family (A n ) of X with x ∈ n A n , there are x n ∈ A n (n ∈ N) with lim n x n = x (see [13]). A topological group G is Fréchet-Urysohn if and only if it is strongly Fréchet-Urysohn (see [1] or [13]).…”

Section: Description Of the Topology Of Cosmic And ℵ 0 -Spacesmentioning

confidence: 99%

“…A topological group G is Fréchet-Urysohn if and only if it is strongly Fréchet-Urysohn (see [1] or [13]). …”

Section: Description Of the Topology Of Cosmic And ℵ 0 -Spacesmentioning

confidence: 99%

On topological properties of Fréchet locally convex spaces with the weak topology

Gabriyelyan

Ka̧kol

Kubzdela

et al. 2015

Topology and its Applications

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Elsevier Gabriyelyan, S.; Kakol, JM.; Kubzdela, A.; López Pellicer, M. (2015). On topological properties of Fréchet locally convex spaces. Topology and its Applications. 192 AbstractWe describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ(E, E ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) 1 and satisfying the Heinrich density condition we prove that (E, σ(E, E )) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain 1 , then (E, σ(E, E )) is an ℵ 0 -space if and only if E is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E )) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF )-space whose strong dual E is separable, then (E, σ(E, E )) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for ǎ Cech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C c (X) is cosmic in the weak topology, (c) the weak * -dual of C c (X) is an ℵ 0 -space.

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“…In [12, p. 798] it is noted also that a Fréchet-Urysohn topological space may not have the weak diagonal sequence property. In [7,Lemma 1.3] it is proved that a Fréchet-Urysohn Hausdorff topological group has the property (AS). It is an open problem, whether the diagonal sequence property is equivalent to the weak diagonal sequence property for topological groups [12] or for topological vector spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Each Fréchet-Urysohn space is sequential and each sequential space has countable tightness (see e.g. [7]). A sequential topological group with the diagonal sequence property is Fréchet-Urysohn [12].…”

Section: Introductionmentioning

confidence: 99%

Diagonal sequence property in Banach spaces with weaker topologies

Plichko

2009

Journal of Mathematical Analysis and Applications

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We investigate the diagonal sequence property in Banach spaces with weaker topologies. In particular, we present examples of Banach spaces with weaker locally convex topologies which have the diagonal sequence property but are not Fréchet-Urysohn. The examples answer negatively a question of Averbukh and Smolyanov. We give also a very simple proof of the fact that each Banach space contains a subset A whose weak closure includes 0, but 0 is not contained in the weak closure of any bounded subset of A.

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A class of angelic sequential non-Fréchet–Urysohn topological groups

Cited by 18 publications

References 13 publications

On topological spaces and topological groups with certain local countable networks

On topological spaces and topological groups with certain local countable networks

On topological properties of Fréchet locally convex spaces with the weak topology

Diagonal sequence property in Banach spaces with weaker topologies

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